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Furkat [3]
3 years ago
11

Find a compact form for generating function of the sequence 4, 4, 4, 4, 1, 0, 1, 0, 1, 0, 1, 0,

Mathematics
1 answer:
Zanzabum3 years ago
5 0

The generating function for this sequence is

f(x)=4+4x+4x^2+4x^3+x^4+x^6+x^8+\cdots

assuming the sequence itself is {4, 4, 4, 4, 1, 0, 1, 0, ...} and the 1-0 pattern repeats forever (as opposes to, say four 4s appearing after every four 1-0 pairs). We can make this simpler by "displacing" the odd-degree terms and considering instead the generating function,

f(x)=3+4x+3x^2+4x^3+\underbrace{(1+x^2+x^4+x^6+x^8+\cdots)}_{g(x)}

where the coefficients of g(x) follow a much more obvious pattern of alternating 1s and 0s. Let

g(x)=\displaystyle\sum_{n=0}^\infty a_nx^n

where a_n is recursively given by

\begin{cases}a_0=1\\a_1=0\\a_{n+2}=a_n&\text{for }n\ge0\end{cases}

and explicitly by

a_n=\dfrac{1+(-1)^n}2

so that

g(x)=\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n

and so

\boxed{f(x)=3+4x+3x^2+4x^3+\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n}

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